Indeed; they have their own bishops and liturgy but acknowledge the authority of the pope. At least in the United States, they tend to be regional in areas where there were many immigrants from the original countries. For example, there's a large Maronite community in Saint Louis and the mayor is Maronite; in Pennsylvania coal country, there's many Byzantine churches founded by immigrants from Eastern Europe. The Maronites are probably closest to Roman Catholicism because they were never officially separated; in other cases, there were often concessions involved. For example, the Byzantines acknowledge the Filioque but don't have to include it in their prayers.
One thing that often is forgotten is that there's different rites even among the parts of the Catholic Church that are usually considered Roman. Up until the Council of Trent, there were often different rites in different regions; the Council standardized the mass but allowed some of the oldest ones such as the Ambrosian Rite to remain. There's even very recent precedent in the Pastoral Provision to assist Anglicans who want to rejoin the Roman Catholic Church en masse (a lot of traditional Anglicans have been converting recently); for example, it gives a dispensation for married Anglican priests to be ordained, and there's parishes and I believe a diocese for them.
With regards to the OP, maybe there could be something similar to the Oxford Movement in the United States, and a backlash against it that drives many Anglicans/Episcopalians into the Roman Catholic Church? I'm not familiar enough with American Anglicanism/Episcopalianism to say if the conditions would work for that.
Thank you for clarifying some aspects. IMHO At the end of the day the Anti-Catholic sentiment in the US is mostly related to the Anti-Irish sentiment. Probably less immigration and no potato famine?